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My 7-year old daughter (pre-school) asked why subtraction is not commutative. How to explain that in simple way, in relation to real world concepts? (I am not looking for abstract mathematical proof)

It is easy to explain why addition is commutative. It is also easy to give real world examples of non-commutative operations (putting on underwear and trousers, etc).

But I cannot figure out example with subtraction.

Bartosz Bilicki
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  • Is she familiar with negative numbers? Or not yet? – Frostic Apr 26 '18 at 17:54
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    If you start with seven apples and give your friend five apples this is not the same as... – Winston Apr 26 '18 at 17:54
  • @MaxFt she understand that you could owe somebody (money, apples) and that this debt is negative number. – Bartosz Bilicki Apr 26 '18 at 17:55
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    @BartoszBilicki I see then negative number might not be the best. I found this article very detailed for teaching purpose. – Frostic Apr 26 '18 at 18:05
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    At her level, it might make more sense to phrase it as "swapping the terms in a subtraction doesn't make sense" rather than "swapping the terms in a subtraction gives a different result". – Jack M Apr 26 '18 at 18:20
  • @JackM thats right. As I see in real world examples second element in subtraction is 'different' than first one. Second is what is taken away, first is what you already have. – Bartosz Bilicki Apr 26 '18 at 18:23
  • (The word in English is subtraction, not “substraction”.) – Hans Lundmark Apr 26 '18 at 20:47

4 Answers4

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The idea that it isn't commutative seems, to me at any rate, to be more intuitive than the idea that it is. Try this: If you laid out 5 coins on the table, you can take away 2, but if you laid out 2 coins on the table, you can't take away 5!

EDIT: Also, if she understands negative numbers, you can explain it using that concept as well (e.g. I can gave you 7 dollars, and you can give me 5. what would it mean if I gave you 5 and you gave me 7?)

Invariance
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Sounds to me like she's smart enough to understand that addition is commutative. So, for example, $$7 - 4 = 7 + (-4).$$ Then $$7 + (-4) = (-4) + 7.$$ But $$4 - 7 = 4 + (-7)$$ and $$4 + (-7) \neq7 + (-4).$$

Robert Soupe
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To make things simple, let's consider a way to make sense of $$1-0 = 1 \ne -1 = 0-1$$ in real life, say the temperature.

  • $1-0 = 1$: yesterday's temperature way $1\mathrm{°C}$, and there's a $0\mathrm{°C}$ drop in temperature today, so the temperature now is $1\mathrm{°C}-0\mathrm{°C} = 1\mathrm{°C}$.
  • $0-1 = -1$: yesterday's temperature way $0\mathrm{°C}$, and there's a $1\mathrm{°C}$ drop in temperature today, so the temperature now is $0\mathrm{°C}-1\mathrm{°C} = -1\mathrm{°C}$.

Both cases give different temperatures.

GNUSupporter 8964民主女神 地下教會
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Some more real-live examples of substractions. They make it easier to understand why substraction is not commutative

  • bank account operations (deposit, withdrawals)
  • sports/games when you can have negative score
  • elevator (negative floor numbers are nowadays common)
Bartosz Bilicki
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